To relax the approximation v1 0 , we use

Prove its validity.

Bernoullis Equation for Fluid Flow Measurement

Equations are based on the frictionless form of Bernoullis equation (F=0).Friction effects are taken into account by introducing empirical coefficients.

Pitot Tube for Open Channel

Point 2 is a stagnation point.In the Pitot tube, the fluid is static.

F is negligible (less than 1%).

v1 2 g h 1

Pitot Static Tube

Neglecting F, we havev1 2

Pitot tubes are used to measure local velocities.

To measure flow rates, the venturi meter and orifice meter are used.

Venturi Meter

Volumetric flow rate = v 2 A 2

From mass balance:v1 v 2

A2A1

Frictionless form of Bernoullis equation ( C v =1):

v2 Cv

2 p1 p 2

A 1 2

A1

in which C v is the discharge coefficient, taking into account friction effects and thefact that flow is not uniform across any section.C v can be estimated knowing the value of Reynolds number (Fig. 5.11):Re1

v 1 D1

The solution procedure is a trial-and-error one.

Inclined Venturi Meter

The venturi meter causes little pressure loss but it is expensive. It is used for large

volumetric flow rates.

p 1 p 2 1 g z 1 z 2 2 1 g z 3 z 4

Orifice MeterIt is simpler than the Venturi meter, but causes more pressure drop. It is used forsmall-size lines.

v2 Cv

2 p1 p 2

A 1 2

A1

C v can be estimated knowing the values of the ratio D 2 / D1 and Reynolds number

(Fig. 5.14): Re 2

v2 D2

RotametersVolumetric flow rate = v 2 A 2 ( z)Approximations:

F is neglected.

p1 and p 3 are assumed uniform.

p2 p3

Result (see derivation in your textbook and in the momentum balance chapter): Fv 2 BF

1/ 2

Rotameters are treated as calibrated devices: using a calibration curve, the flow rate isobtained knowing the float position reading.

Negative Absolute Pressures, Cavitation

Gasesp < 0 has no physical meaning. Velocities are too high in this case.Liquidsp < 0 : flow is unreal physically in this case. There is two-phase flow with higher Fand lower Q.Example 5.12As z increases, p decreases, which can cause boiling if p becomes less than the vaporpressure.Example 5.13As v increases, p decreases, which can cause boiling if p becomes less than the vaporpressure.Bubble flows can cause damage in pumps, turbines, and ships propellers.This phenomenon of local boiling is called cavitation.Unsteady FlowsBernoullis equation can be applied if v max

gx , y ,z

or

1 dpif acceleration is due to pressure. dL

Bernoullis equation can not be applied if the flow is suddenly opened or suddenly

stopped (

v (large in fact)).t

Example 5.14Torricellis equation:v2 2 g h

Mass balance:

A1

dh A 2 2 g hdt

2 h 11 / 2 h 12/ 2

A 2 / A1

2g

In Example 5.14, max

AvA g 2 g for 2 1 (requirement for small v1 andA1tA1

the quasi-steady-state approximation)

Non Uniform FlowsExample: flow over a weirIn this flow, we can not assume the velocity to be practically uniform.Bernouillis equation: